Solve the following undefined questions. a. 2y / (6x + 4)(3x - 9) b. 20 / (x2 + 5x + 6)

Part 2: Solve the right triangle using trigonometry (adjacent side = 5 cm, $\theta = 20^\circ$, opposite = $x$ cm, hypotenuse = $y$ cm)

Step 1: Use tangent for opposite side.
$\tan \theta = \frac{x}{5}$
$\tan 20^\circ = \frac{x}{5}$
$x = 5 \tan 20^\circ \approx 5 \times 0.3640 = 1.82\ \mathrm{cm}$

Step 2: Use cosine for hypotenuse.
$\cos \theta = \frac{5}{y}$
$y = \frac{5}{\cos 20^\circ}$
$y \approx \frac{5}{0.9397} \approx 5.32\ \mathrm{cm}$

Step 3: Verify with sine.
$\sin 20^\circ = \frac{x}{y} \approx \frac{1.82}{5.32} \approx 0.3420$
Matches $\sin 20^\circ$.

Part 3: Solve the triangle using law of cosines (sides $a=6$ cm, $b=10$ cm, $\angle C=10^\circ$, find side $c$, then angles)

Step 1: Law of cosines for side $c$.
$c^2 = a^2 + b^2 - 2ab \cos C$
$c^2 = 6^2 + 10^2 - 2 \cdot 6 \cdot 10 \cdot \cos 10^\circ$
$c^2 = 36 + 100 - 120 \cdot 0.9848 \approx 136 - 118.18 = 17.82$
$c \approx \sqrt{17.82} \approx 4.22\ \mathrm{cm}$

Step 2: Law of cosines for $\angle A$.
$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
$\cos A = \frac{10^2 + 4.22^2 - 6^2}{2 \cdot 10 \cdot 4.22} \approx \frac{100 + 17.81 - 36}{84.4} \approx \frac{81.81}{84.4} \approx 0.9695$
$\angle A \approx \cos^{-1}(0.9695) \approx 14.1^\circ$

Step 3: Find $\angle B = 180^\circ - 10^\circ - 14.1^\circ = 155.9^\circ$.

Part 4: Solve $x = \frac{y + 5}{s}$ for $s$ in terms of $x$ and $y$

Step 1: Multiply both sides by $s$.
$x s = y + 5$

Step 2: Isolate $s$.
$s = \frac{y + 5}{x}$

Part 5: Solve $\frac{dx}{dt} = \frac{dy}{dt} + 10$ for $y' = \frac{dy}{dt}$ (speeds in m/s)

Step 1: Subtract $\frac{dy}{dt}$ from both sides.
$\frac{dx}{dt} - \frac{dy}{dt} = 10$

Step 2: Isolate $y'$.
$y' = \frac{dx}{dt} - 10$

Part 6: Measures of central tendency for data (value, frequency): (2, 6), (3, 10)

Step 1: Mode is the value with highest frequency.
Mode = 3 (frequency 10 > 6)

Step 2: Mean $\bar{x} = \frac{\sum (x \cdot f)}{\sum f}$.
$\sum (x \cdot f) = 2 \cdot 6 + 3 \cdot 10 = 12 + 30 = 42$
$\sum f = 6 + 10 = 16$
$\bar{x} = \frac{42}{16} = \frac{21}{8} = 2.625$
Mean = 2.625

x \approx 1.82 cm
y \approx 5.32 cm
c \approx 4.22 cm
\angle A \approx 14.1^\circ
\angle B \approx 155.9^\circ
s = \frac{y+5{x}
y' = \frac{dx}{dt} - 10
mode = 3
mean = 2.625}